Explanation Given: The function f ( x ) is, f ( x ) = x − 2 The value of a is equal to 1 . Concept used: The n th Taylor polynomial T n forfunction f that is centered at x = a is, T n ( x ) = f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n Here, f ( n ) is the n th derivative of the function f ( x ) . Calculation: The given function f ( x ) is, f ( x ) = x − 2 It is given that the value of a is 1 . The n th Taylor polynomial T n centered at x = 1 is, T n ( x ) = f ( 1 ) + f ′ ( 1 ) 1 ! ( x − 1 ) + f ″ ( 1 ) 2 ! ( x − 1 ) 2 + f ‴ ( 1 ) 3 ! ( x − 1 ) 3 + ⋯ + f ( n ) ( 1 ) n ! ( x − 1 ) n = f ( 1 ) + f ′ ( 1 ) 1 ( x − 1 ) + f ″ ( 1 ) 2 ⋅ 1 ( x − 1 ) 2 + f ‴ ( 1 ) 3 ⋅ 2 ( x − 1 ) 3 + ⋯ + f ( n ) ( 1 ) n ! ( x − 1 ) n Simplify the above n th Taylor polynomial as, T n ( x ) = f ( 1 ) + ( x − 1 ) f ′ ( 1 ) + ( x − 1 ) 2 f ″ ( 1 ) 2 + ( x − 1 ) 3 f ‴ ( 1 ) 6 + ⋯ + f ( n ) ( 1 ) n ! ( x − 1 ) n ....... (1) Substitute 1 for x in the function f ( x ) to obtain the value of f ( 1 ) as, f ( 1 ) = 1 − 2 = 1 1 2 = 1 Differentiate the function f ( x ) with respect to x as, f ′ ( x ) = d d x ( x − 2 ) = − 2 ⋅ x − 2 − 1 = − 2 x − 3 Substitute 1 for x in above equation to obtain the value of f ′ ( 1 ) as, f ′ ( 1 ) = − 2 ( 1 ) − 3 = − 2 ⋅ 1 = − 2 Differentiate the function f ′ ( x ) with respect to x as, f ″ ( x ) = d d x ( − 2 x − 3 ) = − 2 ⋅ d d x ( x − 3 ) = − 2 ( − 3 x − 3 − 1 ) = 6 x − 4 Substitute 1 for

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

See similar textbooks

Question

Chapter 11.7, Problem 27E

To determine

To find: The nth Taylor polynomial Tn for the function f(x)=x−2 centered at x=a for all n .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.7, Problem 26E

Chapter 11.7, Problem 28E

Students have asked these similar questions

A small company of science writers found that its rate of profit (in thousands of dollars) after t years of operation is given by P'(t) = (5t + 15) (t² + 6t+9) ³. (a) Find the total profit in the first three years. (b) Find the profit in the sixth year of operation. (c) What is happening to the annual profit over the long run? (a) The total profit in the first three years is $ (Round to the nearest dollar as needed.)

Find the area between the curves. x= -2, x = 7, y=2x² +3, y=0 Set up the integral (or integrals) needed to compute this area. Use the smallest possible number of integrals. Select the correct choice below and fill in the answer boxes to complete your choice. A. 7 [[2x² +3] dx -2 B. [[ ] dx+ -2 7 S [ ] dx The area between the curves is (Simplify your answer.)

The rate at which a substance grows is given by R'(x) = 105e0.3x, where x is the time (in days). What is the total accumulated growth during the first 2.5 days? Set up the definite integral that determines the accumulated growth during the first 2.5 days. 2.5 Growth = (105e0.3x) dx 0 (Type exact answers in terms of e.) Evaluate the definite integral. Growth= (Do not round until the final answer. Then round to one decimal place as needed.)

Chapter 11 Solutions

EBK CALCULUS

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

Knowledge Booster

The n th Taylor polynomial T n for the function f ( x ) = x − 2 centered at x = a for all n .

Solution Summary: The author explains the nth Taylor polynomial, T_n, for the function f(x) that is centered at x=a

To find: The nth Taylor polynomial Tn for the function f(x)=x−2 centered at x=a for all n .

Want to see the full answer?

Chapter 11 Solutions